A Note on the Acquaintance Time of Random Graphs

Abstract

In this short note, we prove the conjecture of Benjamini, Shinkar, and Tsur on the acquaintance time $\mathcal{AC}(G)$ of a random graph $G \in G(n,p)$. It is shown that asymptotically almost surely $\mathcal{AC}(G) = O(\log n / p)$ for $G \in G(n,p)$, provided that $pn > (1+\epsilon) \log n$ for some $\epsilon > 0$ (slightly above the threshold for connectivity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asymptotically almost surely $K_n$ cannot be covered with $o(\log n / p)$ copies of a random graph $G \in G(n,p)$, provided that $pn > n^{1/2+\epsilon}$ and $p < 1-\epsilon$ for some $\epsilon>0$. We conclude the paper with a small improvement on the general upper bound showing that for any $n$-vertex graph $G$, we have $\mathcal{AC}(G) = O(n^2/\log n )$.